Binary Cubic Groups
On this page:
- Q_{8} - the quaternion group
- 2T - the binary tetrahedral group
- 2O - the binary octahedral group
- 2I - the binary icosahedral group (unfinished)
See also the binary dihedral groups, or dicyclic groups.
We realize these groups as subgroups of the group Sp(1) of unit quaternions. Given a unit quaternion q, the map \(a\mapsto qa\bar q\) (where a is an imaginary quaternion) defines an element of SO(3), giving the well-known double cover Sp(1) → SO(3). (In particular the images of q and -q coincide.)
The binary cubic groups are the preimages under this double cover of the cubic groups T, O, and I. We also include the quaternion group Q_{8}, which is in fact the double cover of the group of order 4 of rotations by \(\pi\) about each of the Cartesian axes. If G is the cubic group in question, one writes 2G for the binary version, and there is a short exact sequence,
$$\mathbb{Z}_2^c \longrightarrow 2G \longrightarrow G $$
where \(\mathbb{Z}_2^c =\{± 1\}\) is the centre of Sp(1). In particular, any representation of G gives rise to a representation of 2G, via the homomorphism 2G → G.
Note that Sp(1) ≅ SU(2).
Q_{8}
\(Q_8\) is the quaternion group and has order 8. It consists of the unit quaternions ±1, ±i, ±j, ±k. It is the double cover of the group D_{2}, which is the subgoup of SO(3) consisting of the identity and the rotation by \(\pi\) about each of the x- y- and z-axes (so \(Q_8\) could be called the binary Klein group). The character table for its real representations is:
Q_{8} | 1 | -1 | {±i} | {±j} | {±k} | notes |
# | 1 | 1 | 2 | 2 | 2 | |Q_{8}|=8 |
A_{0} | 1 | 1 | 1 | 1 | 1 | trivial rep |
A_{1} | 1 | 1 | 1 | -1 | -1 | |
A_{2} | 1 | 1 | -1 | 1 | -1 | |
A_{3} | 1 | 1 | -1 | -1 | 1 | |
H | 4 | -4 | 0 | 0 | 0 | quaternionic type |
- Notice that jij^{-1}=-i, so i and -i are conjugate, as are j and -j, and k and -k.
- The 1-dimensional reps are all induced from the reps of D_{2} = Z_{2}×Z_{2}.
- The irreducible representation H is the natural action of Q_{8} on the space of all quaternions (hence its name). It is therefore of quaternionic type. It is also the underlying representation of the 2-dimensional complex irreducible representation given by Q_{8} ⊂ SU(2) (under the isomorphism of Sp(1) with SU(2)).
- The complex character table has the same 1-dimensional characters, but H should be replaced by E whose character row is (2,-2,0,0,0) = \(\frac12H\).
2T
Here T ≅ A_{4} is the group of all rotational symmetries of the tetrahedron. The binary group 2T is therefore also denoted 2A_{4}. It is (isomorphic to) SL(2; Z_{3}). See Tony Phillips' pages and wikipedia for further details about this group. It contains Q_{8} as a normal subgroup. As unit quaternions, the elements are of the form
±1, ±i, ±j, ±k, ½ (±1 ±i ±j ±k).
The elements of the form ½(1 ±i ±j ±k) are of order 6, while those of the form ½(-1 ±i ±j ±k) are of order 3. And of course ±i etc are of order 4, and -1 is of order 2.
Conjugation within the group permutes the coefficients of the 3 "imaginary" parts and changes an even number of signs, but does not change the real part of a quaternion. For example,
½(1 + i + j + k) is conjugate to ½(1 - i - j + k), and +i is conjugate to -i and to -k.
The conjugacy classes are therefore represented by:
1, -1, i, a = ½(1 + i + j + k), b = ½(1 + i + j - k), c = ½(-1 + i + j + k), d = ½(-1 + i + j - k), so a and b are of order 6 and c, d are of order 3.
Note that a² = d² = c, and b² = c² = d. Finally, Q_{8} is a normal subgroup.
2T | 1 | -1 | i | a | b | c | d | notes |
# | 1 | 1 | 6 | 4 | 4 | 4 | 4 | |2T|=24 |
A_{0} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial rep |
E | 2 | 2 | 2 | -1 | -1 | -1 | -1 | complex type |
T | 3 | 3 | -1 | 0 | 0 | 0 | 0 | |
G | 4 | -4 | 0 | -1 | -1 | 1 | 1 | complex type |
H | 4 | -4 | 0 | 2 | 2 | -2 | -2 | quaternionic type |
- A_{0} E and T arise from reps of T (see A_{4}) via the double covering.
- A_{0} and T are absolutely irreducible (ie, of real type), E and G are of complex type and H is of quaternionic type
- E, G and H split over C, and have characters:
E = (1, 1, 1, ω, ω², ω², ω) ⊕ (1, 1, 1, ω², ω, ω, ω²), where ω = ½(-1+√3 i),
G = (2, -2, 0, ω, ω², -ω², -ω) ⊕ (2, -2, 0, ω², ω, -ω, -ω²), and
H = 2 (2, -2, 0, 1, 1, -1, -1) (2 copies of the same complex rep) - The elements of this group considered as points in the unit sphere in R^{4} (space of quaternions) are the vertices of the convex regular 24-cell. The permutation representation on this set of points is the regular representation, equal to A_{0} + E + 3T +2G + H.
- H is the natural representation of this group acting on the quaternions.
2O
where O is the group of all rotational symmetries of the cube. O is isomorphic to S_{4}, so 2O is often denoted 2S_{4}, it has order 48. See wikipedia for details.
The binary tetrahedral group 2T is a normal subgroup, as is Q_{8.} The remaining elements are all of the form (1/√2)(±1±i) etc.
The conjugacy classes are represented by 1, -1, i, and
2O | 1 | -1 | i | a | c | e | f | g | notes |
# | 1 | 1 | 6 | 8 | 8 | 6 | 6 | 12 | |2O|=48 |
A_{0} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial rep |
A_{1} | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | |
E | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | |
T_{1} | 3 | 3 | -1 | 0 | 0 | 1 | 1 | -1 | |
T_{2} | 3 | 3 | -1 | 0 | 0 | -1 | -1 | 1 | |
H_{1} | 4 | -4 | 0 | 2 | -2 | 2√2 | -2√2 | 0 | action on quaternions |
H_{2} | 4 | -4 | 0 | 2 | -2 | -2√2 | 2√2 | 0 | |
L | 8 | -8 | 0 | -2 | 2 | 0 | 0 | 0 |
- The reps A_{0}, A_{1}, E, T_{1} and T_{2} arise from the corresponding reps of the octahedral group; they are all absolutely irreducible (since they are for S_{4}).
- The reps H_{1}, H_{2} and L are all irreducible of quaternionic type. The corresponding complex irreducible reps have simply half the character.
2I
where I is the group of all rotational symmetries of the icosahedron. 2I has order 120.
I is isomorphic to A_{5}, so 2I is often denoted 2A_{5} (a double cover of A_{5}); it is isomorphic to SL(2,F_{5}). See wikipedia for details.
As unit quaternions, the elements of 2I are:
1, -1, \(\pm i, \, \pm j, \, \pm k\) and \(\frac12(\pm 1 \pm i \pm j \pm k)\) (16 of these) - these form the 24 Hurwitz units, and the subgroup 2T.
and then the 96 quaternions similar to \(\frac12(\pm 0 \pm i\pm \varphi^-j\pm\varphi^+ k)\) formed from even permutations of \(0,1,\varphi^-,\varphi^+\) and all combinations of signs, where \(\varphi^+=\frac12(1+\sqrt5)\) (the golden ratio) and \(\varphi^-=\frac12(1-\sqrt5)\).
Note that \(\varphi^+>0\) while \(\varphi^-<0\). Indeed, in descending order, $$\cos(\pi/5) = \tfrac12 \varphi^+,\quad \cos(2\pi/5) = -\tfrac12 \varphi^-,\quad \cos(3\pi/5) = \tfrac12 \varphi^-,\quad \cos(4\pi/5) = -\tfrac12 \varphi^+$$
Conjugacy classes:
These are classified entirely by the real part of the quaternion:
name | Re(q) | # | order | A_{5} |
1 | 1 | 1 | 1 | e |
-1 | -1 | 1 | 2 | e |
a | 0 | 30 | 4 | C_{2}C_{2} |
b | \(-\frac12\) | 20 | 3 | C_{3} |
c | \(\frac12\) | 20 | 6 | C_{3} |
d | \(-\frac12\varphi^+\) | 12 | 5 | C_{5} |
e | \(-\frac12\varphi^-\) | 12 | 5 | C_{5}' |
f | \(\frac12\varphi^+\) | 12 | 10 | C_{5} |
g | \(\frac12\varphi^-\) | 12 | 10 | C_{5}' |
The column marked A_{5} is the corresponding conjugacy class in A_{5} of the image of the given conjugacy class under the projection (and C_{5} and C_{5}' denote the two different conjugacy classes of elements of order 5 in A_{5}).
Note that the conjugacy classes satisfy \(a^2=-1\), \(b^2=b\), \(c^2=b\), \(d^2=e\), \(e^2=d\), \(f^2=e\) and \(g^2=d\).
Complex representation table:
( need to do real one)
2I | 1 | -1 | a | b | c | d | e | f | g | notes |
# | 1 | 1 | 30 | 20 | 20 | 12 | 12 | 12 | 12 | |2I|=120 |
A_{0} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial rep |
E_{1} | 2 | -2 | 0 | -1 | 1 | \(-\varphi^+\) | \(-\varphi^- \) | \(\varphi^+ \) | \(\varphi^-\) | |
E_{2} | 2 | -2 | 0 | -1 | 1 | \(-\varphi^- \) | \(-\varphi^+\) | \(\varphi^-\) | \(\varphi^+ \) | |
T_{1} | 3 | 3 | -1 | 0 | 0 | \(\varphi^-\) | \(\varphi^+ \) | \(\varphi^+ \) | \(\varphi^-\) | lifted from A_{5} |
T_{2} | 3 | 3 | -1 | 0 | 0 | \(\varphi^+ \) | \(\varphi^-\) | \(\varphi^-\) | \(\varphi^+ \) | lifted from A_{5} |
G_{1} | 4 | 4 | 0 | 1 | 1 | -1 | -1 | -1 | -1 | lifted from A_{5} |
G_{2} | 4 | -4 | 0 | 1 | -1 | -1 | -1 | 1 | 1 | |
H | 5 | 5 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | lifted from A_{5} |
J | 6 | -6 | 0 | 0 | 0 | 1 | 1 | -1 | -1 |
- \(\varphi^+ =\frac12(1+\sqrt5)\) (the golden ratio)
- \(\varphi^-=\frac12(1-\sqrt5)\) (the `rational conjugate' \(=-(\varphi^+)^{-1}\))
- The rational irreducible representations of 2I are \(A_0, E_1+E_2, T_1+T_2, G_1,G_2,H\) and \(J\).
- The labels are not standard.
See also https://people.maths.bris.ac.uk/~matyd/GroupNames/97/SL(2,5).html